Is there a way to convert the following to a linear formulation? In other words, is there a workaround for the absolute value in the objective function?
Minimise: $\left|\textbf{$c^Tx$}\right|=\left|\sum_{i=1}^nc_ix_i\right|$
subject to:
$x_i\in\{1,-1\}$ for all $i=1,...,n$
$c_i\in\mathbb{N}$ for all $i=1,...,n$
I know that if the objective function was $\sum_{i=1}^n|c_ix_i|$ there we could replace each $|x_i|$ with a difference of two positive variables (and I think add some constraints). Is there a similar technique here? Or perhaps a not-so-similar technique?